Question

If $A$ and $B$ are square matrices such that $A^{\ast }=A$ and $B^{\ast }=B$, where $A^{\ast }$ denotes the conjugate transpose of $A$, then ${(AB-BA)}^{\ast }=$

• A. Null matrix
• B. $AB-BA$
• C. $BA-AB$
• D. none of these

Answer 1

The correct option is C $BA-AB$

${(AB-BA)}^{\ast }=\overline{{(AB-BA)}^{\prime }}$
$=\overline{(AB{)}^{\prime }-(BA{)}^{\prime }}$
$=\overline{(B^{\prime }{A}^{\prime }-A^{\prime }{B}^{\prime })}$
$=\overline{B^{\prime }{A}^{\prime }}-\overline{A^{\prime }{B}^{\prime }}$
$=\overline{B^{\prime }}\overline{A^{\prime }}-\overline{A^{\prime }}\overline{B^{\prime }}$
$=B^{\ast }{A}^{\ast }-A^{\ast }{B}^{\ast }$
$=BA-AB$